Question

Ramdeen gives $40\% $ of his total property to his son. He gives $60\% $ of the balance to his daughter and the remaining sum to his wife. If the wife gets Rs. 1920 less than his daughter, then find the value of his property.

Answer 1

Hint:
We are required to find the total property of Ramdeen. To do that we will assume his property as an unknown variable. We will then form the various linear equations based upon the given conditions, and solve them to find the total property of Ramdeen.

Complete step by step solution:
We are required to find the total property of Ramdeen in this question. To do that we will proceed as follows –
Let us assume that the total property of Ramdeen is $x$.
According to the question Ramdeen gives $40\% $ of his total property to his son. This means that,
Property received by Ramdeen’s son = $40\% $ of Ramdeen’s total property
Property received by Ramdeen’s son = $40\% $ of $x$
Property received by Ramdeen’s son = $\dfrac{{40}}{{100}} \times x$
Hence, the remaining property of Ramdeen now will be,
${\text{Remaining Property}} = {\text{Total property}} - {\text{Property received by Ramdeen's son}} \\
   = x - \dfrac{{40x}}{{100}} \\
   = \dfrac{{100x}}{{100}} - \dfrac{{40x}}{{100}} \\
   = \dfrac{{60x}}{{100}} \\$
Now, according to the question, Ramdeen gives $60\% $ of the balance to his daughter. This means that he gives $60\% $ of the remaining property to his daughter.
So, now we will calculate the property received by Ramdeen’s daughter.
${\text{Property received by Ramdeen's daughter}} = 60\% {\text{ of Remaining property}} \\
   = 60\% \times \dfrac{{60x}}{{100}} \\
   = \dfrac{{60}}{{100}} \times \dfrac{{60x}}{{100}} \\
   = \dfrac{{36}}{{100}}x \\$
Also, according to the question, Ramdeen gives $60\% $ of the balance to his daughter and the remaining sum to his wife. This means that after giving $60\% $ of the remaining property to his daughter, he gave the rest of the remaining property to his wife.
So, now we will calculate the property received by Ramdeen’s wife.
${\text{Property received by Ramdeen's Wife}} = {\text{Remaining property}} - {\text{Property received by Ramdeen's daughter}} \\
   = \dfrac{{60x}}{{100}} - \dfrac{{36x}}{{100}} \\
   = \dfrac{{24}}{{100}}x \\$
Now the question also says that Ramdeen’s wife gets Rs. 1920 less than his daughter.
So, we will write the equation for the above statement as,
${\text{Property received by Ramdeen's daughter}} - {\text{Property received by Ramdeen's daughter}} = {\text{Rs }}1920 \\
  \dfrac{{36x}}{{100}} - \dfrac{{24x}}{{100}} = 1920 \\
  \dfrac{{12}}{{100}}x = 1920 \\$
Now we will solve the above linear equation to find the value of $x$. To do that, we will multiply both sides of the equation by 100.
$\dfrac{{12x}}{{100}} = 1920 \\
   \Rightarrow \dfrac{{12x}}{{100}} \times 100 = 1920 \times 100 \\$
Now we will divide both sides of the equation with 12.
$\Rightarrow 12x = 192000 \\
   \Rightarrow \dfrac{{12x}}{{12}} = \dfrac{{192000}}{{12}} \\$
Now we will divide the numerator from the denominator.
$
   \Rightarrow x = \dfrac{{192000}}{{12}} \\
   \Rightarrow x = 16000 \\
$

$\therefore $ The total property of Ramdeen is Rs. 16000.

Note:
The question says that ‘Ramdeen gives $40\% $ of his total property to his son. He gives $60\% $ of the balance to his daughter’. Here, we might think that he divides his property amongst his son and daughter in the ratio of 40 and 60. We might infer that after giving the property to his son he gives all the remaining property to his daughter. However, this will be wrong. The statement ‘He gives $60\% $ of the balance to his daughter’ means that after giving the property to his son, whatever is left, Ramdeen gives the $60\% $ of that remaining property to his daughter.